An integrable model of a planar tri-atomic molecule.

This article presents an integrable model of a planar tri-atomic molecule within the Born–Oppenheimer adiabatic approximation, where the total molecular Hamiltonian is composed of a nuclear vibrational energy operator and an electronic Hamiltonian defined as a traceless 2 × 2 Hermitian matrix on the nuclear shape space. The study introduces vibrational-electronic (vibronic) interaction via covariant differential operators acting on sections of eigen-line bundles associated with chosen electronic eigenvalues, leading to modified nuclear kinetic energy operators that incorporate geometric (Berry) phases manifested as monopole bundles over the shape space. The eigenvalue problem for nuclear motion coupled with electronic states is solved explicitly, revealing bound states with energy levels shifted by topological invariants known as Chern numbers, and showing that the positive electronic eigenvalue yields bound molecular states while the negative one leads to dissociation. The model generalizes known results on the dynamic Jahn–Teller effect for small distortions around symmetric configurations and relates the integrability of the system to that of a four-dimensional harmonic oscillator reduced by symmetry considerations.